Circuitry that implements printed circuit boards (PCBs), very large scale integrated systems (VLSI), etc., often includes a strip of metal (also called a trace) disposed between two conductive layers/planes, (e.g., metallic layers/planes). For example, as shown in FIG. 1, one of the conductive layers/planes is called a power plane and the other layer/plane is called a ground plane. Often a substrate layer (e.g., a dielectric layer) separates the power plane from the trace and another substrate layer (e.g., another dielectric layer) separates the ground plane from the trace. Usually, a same material with a same composition is used to form both substrate layers. In determining electrical properties of the circuitry, electrical properties of such a layered structure including the power plane, the ground plane and the trace disposed in between are analyzed.
A modal decomposition technique is commonly applied in which transmission line modes are separated from parallel plate modes to analyze the electrical properties of the layered structure as shown in FIG. 1. An electromagnetic (EM) field between planes (e.g., a parallel plate mode) is modeled using a 2D full-wave solver in order to accommodate various geometries, specifically, power planes having non-rectangular shapes. In such modal decomposition solvers, a transmission line mode is represented by a two-port modal admittance matrix:
      Y    port    =      [                                                      Z                              -                1                                      ⁢            T            ⁢                                                  ⁢                          Λcoth              ⁡                              (                                  Λ                  ⁢                                                                          ⁢                  l                                )                                      ⁢                          T                              -                1                                                                                        -                              Z                                  -                  1                                                      ⁢            T            ⁢                                                  ⁢                          Λcsch              ⁡                              (                                  Λ                  ⁢                                                                          ⁢                  l                                )                                      ⁢                          T                              -                1                                                                                                    -                              Z                                  -                  1                                                      ⁢            T            ⁢                                                  ⁢                          Λcsch              ⁡                              (                                  Λ                  ⁢                                                                          ⁢                  l                                )                                      ⁢                          T                              -                1                                                                                        Z                              -                1                                      ⁢            T            ⁢                                                  ⁢                          Λcoth              ⁡                              (                                  Λ                  ⁢                                                                          ⁢                  l                                )                                      ⁢                          T                              -                1                                                          ]  The transmission line parameters are extracted using a separate 2D transmission line solver. The coupling between a stripline (i.e., a transmission line) mode and a parallel plate waveguide mode can be analyzed by using a multi-conductor transmission line (MTL) theory in which the power plane is considered to be a wide trace. This classic, widely used modal decomposition technique is restricted, however, to situations in which a homogenous substrate (e.g., a dielectric material) is disposed between the power plane and the trace and between the ground plane and the trace.
In some circuit implementations, instead of substantially the same substrate material separating the trace from both the power plane and the ground plane, one or more dielectric or substrate layers separate the trace from the power plane, and one or more substrate layers separate the trace from the ground plane. The one or more substrates separating the power plane from the trace are different than the one or more substrates separating the ground plane from the trace. The substrates can be different in terms of materials, compositions, etc. The classical modal decomposition technique is generally not applicable to structures using non-homogeneous (also called inhomogeneous) substrates. One reason is, for traces in inhomogeneous material and coplanar waveguides, the classical method utilizing a two-port admittance matrix usually causes parasitic power plane resonances and can lead to an inaccurate transmission line model.
A strict modal decomposition approach may accommodate inhomogeneous substrates (e.g., B J. Yang, M. Swaminathan, “Simple Equivalent Circuit Model of a Stripline in Inhomogeneous Dielectric Media,” IEEE Microwave and Wireless Comp. Letters, Vol. 19, No. 12, p 771, 2009, incorporated by reference in its entirety herein), but this technique is limited to rectangular power planes only, and is not applicable for analysis/simulation of the EM field between the power plane and the ground plane that have shapes other than simple rectangles which are increasingly employed in the fabrication of circuitry. In another approach (e.g., E. Engin, W. John, G. Sommer, W. Mathis, H. Reichl, “Modeling of Striplines Between a Power and Ground Plane,” IEEE Trans. Adv. Packag., vol. 29, no. 3, p 415, 2006, incorporated by reference in its entirety herein), correction terms are added to the modal decomposition to account for inhomogeneity, but significant inaccuracies may be present in the simulation/analysis with the addition of the correction terms.